Systems and methods for minimizing aberrating effects in imaging systems

ABSTRACT

An imaging system for reducing aberrations from an intervening medium, and an associated method of use are provided. The system may be an optical or task-based optical imaging system including optics, such as a phase mask, for imaging a wavefront of the system to an intermediate image and modifying phase of the wavefront such that an optical transfer function of the system is substantially invariant to focus-related aberrations from the medium. A detector detects the intermediate image, which is further processed by a decoder, removing phase effects from the optics and forming a final image substantially clear of the aberrations. Other systems may employ an encoder that codes wavefronts of acoustical waves propagating through a medium to make the wavefronts substantially invariant to acoustical aberrations from the medium. Imaging and decoding of the wavefronts reverse effects of the wavefront coding and produce sounds substantially free of the aberrations.

RELATED APPLICATIONS

This application is a divisional application of U.S. patent applicationSer. No. 10/813,993, filed Mar. 31, 2004 which claims priority to U.S.Provisional Application Ser. No. 60/459,417, filed Mar. 31, 2003. Theaforementioned applications are incorporated herein by reference.

BACKGROUND

One goal of an optical imaging system design is to capture nearlyerror-free images. The optical design thus specifically seeks to correctfor certain known optical influences, including, for example, aberratingeffects of a medium through which images are captured and unwantedreflections (e.g., scattering) within the imaging system.

Compensating for aberrating effects of the medium is often necessarybecause the medium unacceptably distorts the optical wavefront, leadingto degraded images. The Earth's atmosphere is an example of one mediumthat can create such degraded images. Turbulent water is another exampleof such a medium. The only medium that does not affect the opticalwavefront is a vacuum at zero atmosphere, which is idealized andpractically unachievable.

The prior art has devised adaptive optics to overcome certain problemsassociated with optical distortions induced by the medium. In typicalprior art systems incorporating adaptive optics, information about themedium-induced aberrations is first obtained. After the information isacquired, it is then used to modify or “adapt” the optics of the opticalimaging system so as to compensate for the aberrations. The ability ofthe adaptive optics to compensate for the aberrations is thus directlyrelated to obtaining accurate information concerning the aberrations asgenerated by the medium.

One prior art technique for obtaining information about aberrationsinduced by the medium requires direct measurement of phase effects of anoptical wavefront traveling through the medium at the aperture stop ofthe optical imaging system. By measuring the phase of the opticalwavefront from a point source with, for example, an interferometer, theoptical wavefront may be corrected by changing or “adapting” an opticalelement, such as a deformable mirror in the optical imaging system.Another term often used to describe adaptive optical elements is“wavefront correction,” which implies that the phase errors of theoptical wavefront are corrected at the aperture stop. Theaberration-induced effects caused by the medium typically change overtime. As the properties of the medium vary, therefore, the point spreadfunction (“PSF”) or spatial impulse response of the optical imagingsystem also varies. Consequently, the adaptive optics must also changewith time, and the phase effects of the optical wavefront must again bedetermined. These requirements lead to a complex process and a highlyinvolved optical imaging system.

Another prior art technique forms an image of a known object todetermine the PSF of the optical imaging system. Typically, this knownobject is a point source such as a guide star (e.g., non-resolvablestar) or a satellite in the field of view of the optical imaging system.Since the PSF is affected by aberrations of the medium, as well as byaberrations specific to the optical imaging system, the PSF may beintegrated over the exposure time to acquire the impulse response ofboth the optical imaging system and the medium. The PSF is then used todeconvolve each subsequent image to obtain a final image that isessentially equivalent to an image that would be obtained if noaberrations were induced by the medium. This technique, however, has asignificant shortcoming due to the requirement of a reference point; forexample a non-resolvable star is not often available near the object ofinterest. In another example, if a satellite serves as a reference, themovement of the satellite makes it difficult to synchronize with primaryimaging. In more practical situations on earth, such as imagingground-based objects with a telescope, there are often no isolated orsuitable point reference objects.

Other prior art methods obtain information about aberrations in a mediumand do not use an image of a non-resolvable point but attempt to extractinformation concerning the object from a series of images, while theproperties of the aberrating medium change over time. These methods,however, produce images with a high level of noise. Furthermore,attempting to remove all time-varying portions of such images in aseries, to obtain a good estimate of the imaged object, requiresconsiderable computing power. In addition, errors are induced when themedium changes and images are taken without the benefit of a currentaberration-removing calculation.

In the prior art, one method to compensate for unwanted reflectionswithin and from an optical imaging system is to strategically employ aprism within the system. However, introducing the prism into the path ofa converging optical wavefront introduces other aberrations. Moreover,the use of a prism within the system only partially compensates for theunwanted reflections and induces thermal and throughput problems.

SUMMARY

Systems and methods are disclosed for reducing the effects ofaberrations in optical imaging systems. In one aspect, an opticalimaging system corrects certain aberrations when imaging through amedium. By coding the optical wavefront imaged onto the system'sdetector, and by post processing data from the detector, the system ismade substantially invariant to such aberrations caused by the mediumthrough which a wavefront passes. The wavefront may be, for example, acommon phase-front of electromagnetic radiation (e.g., visible,infrared, ultraviolet, radio wave, etc.) imaged by the optical imagingsystem. The wavefront may also be a phase front of acoustic waves in anacoustic imaging system. The aberrations are, for example, focus-relatedaberrations like Petzval (field curvature), astigmatism, thermalvariations in the system and/or medium, pressure (ripple) variationswithin the medium, weather-related effects of the medium, etc.

In another aspect, the optical imaging system includes optics that codethe wavefront to correct the effects of the aberrations. Such optics maycomprise a mask (e.g., a phase mask) that modifies the optical transferfunction of the system to account for certain aberrating effects of themedium such as defined by Zernike polynomials. Coding of the wavefrontmay also occur through an aspheric optical element forming one or moresurfaces of the optics.

In one aspect, the medium is air and the wavefront coded optical systemoperates to diminish the effects of refractive index changes in the air(e.g., induced by temperature and/or barometric pressure changes). Sucha system is, for example, useful in lithography.

In another aspect, a decoder performs post processing to generate asubstantially aberration-free final image by removing effects of themask on data from the detector. By way of example, the decoder acts toremove spatial blurring in the image data, caused by the mask, throughconvolution to generate the final image.

In yet another aspect, a low reflectivity optical imaging system isformed with optics that introduce tilt at an aperture stop of the systemto deviate reflected waves such that the waves are blocked by anaperture of the system. Aberrations created by the tilt may be furthercorrected by wavefront coding and post-processing of a detected image toremove the aberrations. Wavefront coding configurations with or withouta tilt at the aperture stop can also be used to further decreaseunwanted reflections while also achieving a large depth of field,aberration tolerance, and/or anti-aliasing.

In still another aspect, wavefront coding optics are used within imagesighting systems to diminish the effects of certain illuminatingsources, such as a laser. In this aspect, the wavefront coding optics(e.g., a phase mask) spatially diffuses the incoming signal from thesource such that it is less damaging to a receiving detector or a humaneye, and/or such that the reflection from such sources are much lowerthan the reflection that would occur without the wavefront codingoptics.

U.S. Pat. No. 5,748,371 is incorporated herein by reference.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a prior art optical imaging system.

FIG. 2 shows one optical imaging system with wavefront coding optics.

FIG. 3 shows a pupil map and corresponding 2D optical modulationtransfer function (“MTF”), sample PSF, and Optical MTF for a segmentedoptical system without piston error.

FIG. 4 shows a pupil map and corresponding 2D optical MTF, sample PSF,and MTF curves for a segmented optical system with one segment pistonerror.

FIG. 5 shows a pupil map and corresponding 2D optical MTF, sample PSF,and optical MTF curves for a segmented optical system with two segmentpiston error.

FIG. 6 shows modulation transfer functions for a representativeconventional optical imaging system, an optical imaging system(employing wavefront coding) before filtering, and an optical imagingsystem (employing wavefront coding) after filtering.

FIG. 7A shows image intensity plots for a representative conventionaloptical imaging system; FIG. 7B shows image intensity plots for anoptical imaging system employing wavefront coding.

FIG. 8 shows pupil maps and associated MTF curves for an adaptiveoptical element showing effects of quilting and stuck actuator errors.

FIG. 9 shows composite pupil maps and illustrative MTF curvesillustrating effects of wavefront coding and post processing on adaptiveoptics affected by quilting and stuck actuator errors.

FIG. 10 shows an adaptive optics pupil overlaid with a phase function ofone phase mask, and an associated Zernike polynomial.

FIG. 11 illustrates certain misfocus effects due to sphericalaberration.

FIG. 12 illustrates certain misfocus effects due to astigmatism.

FIG. 13 illustrates certain misfocus effects due to Petzval curvature.

FIG. 14 illustrates certain misfocus effects due to axial chromaticaberration.

FIG. 15 illustrates certain misfocus effects due to temperature inducedaberrations.

FIG. 16 illustrates certain aberrations due to coma.

FIG. 17 shows a comparison of PSFs between traditional imaging systemsand imaging systems (employing wavefront coding) as to coma effects, asa function of misfocus.

FIG. 17A illustrates the exit pupil optical path difference (OPD) and apolynomial representation.

FIG. 17B shows a comparison of MTF curves over spatial frequency forwaves of misfocus in a diffraction-limited system, and the same systemaffected by trefoil or coma aberrations.

FIG. 17C shows a comparison of MTF curves over spatial frequency forwaves of misfocus in a modified diffraction-limited system, and the samesystem affected by trefoil or coma aberrations, and the same systemsafter linear filtering.

FIG. 17D shows a comparison of PSFs for several waves of misfocus formodified and unmodified diffraction-limited systems with trefoil andcoma aberrations, with linear filtering.

FIG. 18 shows a thermal sighting system with an optical element(employing wavefront coding) to diffuse incoming radiation and reducereflections.

FIG. 19 shows two phase forms for use with the element of FIG. 18.

FIG. 20A shows a prior art low reflectivity optical imaging system; FIG.20B shows a ray intercept map of the optical imaging system of FIG. 20A.

FIG. 21A shows a low reflectivity optical imaging system with a tiltelement; FIG. 21B shows a ray intercept map of the optical imagingsystem of FIG. 21A.

FIG. 22 shows intensity profiles for reflected ray energy for aconventional optical imaging system and two optical imaging systemsemploying wavefront coding.

FIG. 22A shows a plot of integrated reflected power from a traditionaldiffraction-limited system and an optical imaging system employingwavefront coding.

FIG. 22B shows a plot of MTF curves over normalized spatial frequencyfor a traditional diffraction-limited system and optical imaging systems(employing wavefront coding) with and without filtering.

FIG. 22C illustrates an imaging exit pupil related to the opticalimaging system of FIG. 22A.

FIG. 22D illustrates a mesh view and an image view of a sampled PSFformed from the imaging exit pupil of FIG. 22C.

FIG. 23 shows an imaging system for imaging acoustical waves withwavefront coding.

FIG. 24 shows an optical imaging system with automated softwarefocusing.

FIG. 25 illustrates certain software processing steps for the system ofFIG. 24.

FIG. 26 illustrates certain other software processing steps for thesystem of FIG. 24.

FIG. 27 graphically illustrates focus score for certain exemplary cases.

FIG. 28A shows one task-based optical imaging system.

FIG. 28B shows one task-based optical imaging system employing wavefrontcoding.

FIG. 28C shows one other task-based optical imaging system employingwavefront coding.

FIG. 29 shows one task-based, iris recognition optical imaging systememploying wavefront coding.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 schematically shows a prior art optical imaging system 10 thatimages an object 50 through a medium 52 to a detector 58 (e.g., a CCDarray). Detector 58 senses electromagnetic radiation 54 that emitsand/or reflects from object 50 and that is imaged by optics 56 (e.g.,one or more lenses) to detector 58. The electromagnetic radiation 54imaged at detector 58 is often characterized by an optical wavefront 55,representing a constant phase front of radiation 54. Image processing 60may then process data from detector 58, for example to provide edgesharpening, color filter array interpolation and/or contrast imageadjustment.

An optical imaging system 100 is schematically shown in FIG. 2 tominimize certain aberrations (e.g., misfocus-related aberrations)introduced by medium 52′. Through operation of optics 102 (e.g., one ormore optical lenses or mirrors), imaging system 100 images an object 50′through medium 52′ to a detector 106, which converts focusedelectromagnetic radiation 54′ to output data 111. Detector 106 is forexample a CCD array, a CMOS array, an IR detector such as a bolometer,etc.

In addition to performing imaging functions, optics 102 also encodes anoptical wavefront 55′ from object 50′ with a phase function, describedin more detail below. A decoder 108 processes data 111 from detector 106to produce a final image 110, which is substantially equivalent to animage that would be obtained by detector 106 if no aberrations wereinduced by medium 52′. In one embodiment, decoder 108 operates byreversing certain spatial effects induced by wavefront coding ofwavefront 55′, by optics 102, with the phase function. By way ofillustration, decoder 108 may perform a convolution on data 111 with aconvolution kernel related to the phase function representing one ormore aspherical surfaces within optics 102. Decoder 108 may also act toextract certain information from the detected image. This informationcould, for example, be a code related to an imaged iris, or related to alocation of a detected object. In these examples the final image 110need not be suitable for human viewing but may be suitable forrecognition by a machine.

More particularly, by operation of optics 102, imaging ofelectromagnetic radiation 54′ (reflected and/or emitted by object 50′)to detector 106 does not form a sharp image; rather, the focusedelectromagnetic radiation 54′ at detector 106 is spatially blurred inimaging system 100, as indicated by blur 104. Detector 106 senses thefocused, spatially blurred electromagnetic radiation 54′. Decoder 108thus serves to remove effects of the spatial blurring, such as through aconvolution, utilizing the phase form which initially caused theblurring. By altering the phase front of wavefront 55′, optics 102 thusmodifies the optical transfer function of optical imaging system 100;this optical transfer function is substantially the same for a range offocus positions about a best focus position at detector 106 (the bestfocus position being determined as if optics 102 did not encodewavefront 55′).

In one example, medium 52′ is the Earth's atmosphere. In anotherexample, medium 52′ is turbulent water. Medium 52′ may be any mediumthat transmits electromagnetic radiation 54′, other than an idealizedzero atmosphere vacuum.

Electromagnetic radiation 54′ is, for example, visible radiation,infrared radiation, ultraviolet radiation, radio waves, or any otherportion of the electromagnetic spectrum, or combination thereof.Radiation 54′ may also be acoustic radiation.

To encode wavefront 55′, optics 102 includes a phase mask 103 thatmodifies the phase of wavefront 55′ with the phase function. Mask 103may be a separate optical element, or it may be integral with one ormore optical elements of optics 102; for example, mask 103 may also bemade on one or more surfaces of such optical elements. By way ofillustration, one family of phase functions (each phase functionequivalent to a surface height profile) induced by mask 103 may berepresented by the following:Separable-forms(x,y)=Σa _(i)[sign(x)|x| ^(bi)+sign(y)|y| ^(bi)],where|x|≦1,|y|≦1,andsign(x)=+1 for x≧0, sign(x)=−1 otherwise.Another exemplary family of phase functions may be described as:Non-separable-forms(r,theta)=Σa _(i) r ^(bi) cos (w _(i)theta+phi_(i))where the sum is over the subscript i. Yet another family of phasefunctions is described by constant profile path optics set forth incommonly-owned, pending U.S. application Ser. No. 10/376,924, filed on27 Feb. 2003 and incorporated herein by reference. In practice,different phase functions or different families of phase functions canbe combined to form new wavefront modifying phase functions.

Optics 102 may additionally include one or more adaptive optics 105, toassist in correcting distortions within wave front 55′ due to medium52′. In one embodiment, elements 105 and mask 103 comprise one and thesame optical structure.

One benefit of the phase function applied by mask 103 is that it may bedesigned to absorb little or no energy from electromagnetic radiation54′, obviating the need for increased exposure or illumination and yetmaintaining benefits of minimizing certain aberrating effects of medium52′. In one embodiment, phase mask 103 is located either at or near oneof the following locations within system 100: a principal plane, animage of the principal plane, an aperture stop, an image of the aperturestop, a lens or a mirror.

The aberrating effects induced by medium 52′ may be modeled to optimizeimage processing by system 100, for example to make system 100substantially invariant to focus-related aberrations across a broadspectrum of aberrations. Aberrations introduced by medium 52′ mayinclude, for example, chromatic aberration, curvature of field,spherical aberration, astigmatism, and temperature or pressure relatedmisfocus often associated with plastic or infrared (IR) optics.

To encode the phase function onto wavefront 55′, phase mask 103 may, forexample, have variations in opaqueness, thickness and/or index ofrefraction, which affect the phase of wavefront 55′. Planar or volumeholograms or other phase-changing elements may be used as mask 103. Moreparticularly, errors or aberrations introduced by medium 52′ inwavefront 55′ can be characterized by the following geometric series:Φ=a+bx+cx ² +cx ³+. . . .where the first term represents a constant phase shift, the second termrepresents a tilt of the phase, the third term represents a misfocus,the fourth term represents a cubic error, etc. All terms that have aneven number exponent are focus-related errors, such as chromaticaberration, curvature of field, spherical aberration, astigmatism andtemperature or pressure related misfocus. Through the coding ofwavefront 55′, these focus-related errors introduced by medium 52′ arereduced or minimized within optical imaging system 100. Optics 102 mayfurther include corrections to reduce non-focus related errors, i.e.,the odd number exponent terms in the above geometric series. Such errorsinclude phase shift and comatic errors. All errors or aberrations may becontrolled by combinations of wavefront coding optics 102 and mask 103.

Zernike polynomial analysis may be used to characterize errors oraberrations induced by medium 52′ in wavefront 55′. In determining thesensitivity of optics 102 (and mask 103) to minimize these aberrations,Siedel aberrations may be used. The odd and even Zernike polynomials aregiven by: $\begin{matrix}{\begin{matrix}{{{}_{}^{}{}_{}^{}}\left( {\rho,\phi} \right)} \\{{{}_{}^{}{}_{}^{}}\left( {\rho,\phi} \right)}\end{matrix} = {{R_{n}^{m}(\rho)}\begin{matrix}\sin \\\cos\end{matrix}\left( {m\quad\phi} \right)}} & (1)\end{matrix}$where the radial function R_(n) ^(m)(ρ) is defined for n and m integerswith n≧m≧0 by $\begin{matrix}{{R_{n}^{m}(\rho)} = \left\{ \begin{matrix}\sum\limits_{l = 0}^{{({n - m})}/2} & {\frac{\left( {- 1} \right)^{l}{\left( {n - l} \right)!}}{l\quad{{{{1\left\lbrack {{\frac{1}{2}\left( {n + m} \right)} - l} \right\rbrack}!}\left\lbrack {{\frac{1}{2}\left( {n - m} \right)} - l} \right\rbrack}!}}\rho^{n - {2l}}} & \quad \\\quad & \quad & {{{for}\quad n} - {m\quad{even}}} \\0 & \quad & {{{for}\quad n} - {m\quad{{odd}.}}}\end{matrix} \right.} & (2)\end{matrix}$Here, φ is the azimuthal angle with 0≦φ<2π and ρ is the radial distancewith values between and including 0 and 1. The even and odd polynomialsare sometimes also denoted as: $\begin{matrix}{{Z_{n}^{- m}\left( {\rho,\phi} \right)} = {{{{}_{}^{}{}_{}^{}}\left( {\rho,\phi} \right)} = {{R_{n}^{m}(\rho)}{\sin\left( {m\quad\phi} \right)}}}} & (3) \\{{Z_{n}^{m}\left( {\rho,\phi} \right)} = {{{{}_{}^{}{}_{}^{}}\left( {\rho,\phi} \right)} = {{R_{n}^{m}(\rho)}{{\cos\left( {m\quad\phi} \right)}.}}}} & (4)\end{matrix}$

Table 1 shows the mathematical form of certain representative Zernikeaberrations and whether errors can be corrected using optics 102. TABLE1 # Mathematical Form Aberration Static Errors Dynamic Errors 0 1 PistonCorrectable Correctable 1 ρ cosθ X-tilt NA NA 2 ρ sinθ Y-tilt NA NA 32ρ² − 1 Focus Correctable Correctable 4 ρ² cos2θ Astigmatism CorrectableCorrectable 5 ρ² sin2θ Astigmatism Correctable Correctable (45°) 6 (3ρ²− 2) ρ cosθ Coma Correctable Not Fully Correctable With Fixed LinearFiltering 7 (3ρ² − 2) ρ sinθ Coma Correctable Not Fully Correctable WithFixed Linear Filtering 8 6ρ⁴ − 6ρ² + 1 Spherical Correctable Correctable9 ρ³ cos3θ Trefoil Correctable Not Fully Correctable With Fixed LinearFiltering 10 ρ³ sin3θ Trefoil Correctable Not Fully Correctable WithFixed Linear Filtering 11 (4ρ² − 3)ρ² cos2θ Spherical & CorrectableCorrectable Astigmatism 12 (4ρ² − 3)ρ² sin2θ Spherical & CorrectableCorrectable Astigmatism 13 (4ρ⁴ − 12ρ² + 3) ρ Coma Correctable Not Fullycosθ Correctable With Fixed Linear Filtering 14 (4ρ⁴ − 12ρ² + 3) ComaCorrectable Not Fully ρ sinθ Correctable With Fixed Linear Filtering 1520ρ⁶ − 30ρ⁴ + Spherical Correctable Correctable 12ρ² − 1

Table 1 shows that static errors and dynamic errors can be corrected byoptics 102 (including mask 103) for a number of aberrations. In oneexample, static and dynamic errors brought about by coma aberrations maybe corrected with optics 102. In another example, x-tilt and y-tiltaberrations are not corrected with optics 102 for general unknown tilts,but are corrected by other methods. Coma and Trefoil are specialaberrations that can be corrected although specialized signal processingmay be required.

If medium 52′ is, for example, a turbulent medium such as Earth'satmosphere, adaptive optics 105 may still be used. However, because ofoptics 102 (and mask 103), the amount of aberration correction performedby separate adaptive optics (e.g., adaptive optics 105), if used, isreduced. Accordingly, optical imaging system 100 may be designed tominimize the system's coma and lateral chromatic aberrations, as well asthe probability that coma and lateral chromatic aberrations will arise,through optimization and tolerancing of system 100. More particularly,the combination of optics 102 (with mask 103) and the optimization ofsystem 100 to minimize coma and lateral chromatic aberrations results ina robust imaging system 100 that minimizes aberrating effects introducedby medium 52′.

As described in more detail below, one skilled in the art appreciatesthat adaptive optics 105 may include a segmented mirror, each part ofthe segmented mirror being moveable (or actuated) to adapt the wavefrontto a desired phase form. Those skilled in the art also appreciate thatpiston error may result from such segmented mirrors. Fortunately, pistonerror is one form of aberration that may also be minimized by opticalimaging system 100 with optics 102.

FIG. 3 illustrates a pupil function 105A illustrating segmented adaptiveoptics. Pupil function 105A has an obscuration 106A at its center and isfree of piston error across each segment. The optical modulationtransfer function (“optical MTF”, or “MTF”) of a wavefront passingthrough pupil function 105A is illustratively shown in graph 110A. Graph110A shows a two-dimensional (2D) MTF for pupil function 105A,illustrating diffraction-limited performance. Graph 112A specificallyshows traces along the vertical (y) and horizontal (x) axes for the 2DMTF, which has a steady but decreasing modulation as spatial frequencyincreases. A sampled point spread function (“PSF”) as sampled by adetector is shown in graph 114A.

FIG. 4 illustrates a pupil function 105B illustrating another segmentedadaptive optics. Pupil function 105B has an obscuration 106B at itscenter and has one segment 107B with a pi/2 phase shift piston error.The MTF of a wavefront passing through pupil function 105B isillustratively shown in graph 110B. Graph 110B shows the 2D MTF forpupil function 105B, illustrating less than diffraction-limitedperformance due to a phase shift of segment 107B. Graph 112Bspecifically shows a reduction in contrast, as compared to graph 112A,as spatial frequency increases. A sampled point spread function (“PSF”)as sampled by a detector is shown in graph 114B. The sampled PSF ofgraph 114B noticeably broadens (as compared to the PSF of graph 114A)due to a reduction in spatial resolution.

FIG. 5 is similar to FIG. 4, and illustrates the same segmented opticaladaptive optics but with two segment piston error shown in a pupilfunction 105C. Pupil function 105C has two segments 107C, each having api/2 phase shift piston error. Pupil function 105C has a centralobscuration 106C, as above. The MTF of a wavefront passing through pupilfunction 105C is illustratively shown in graph 110C. Graph 110C showsthe 2D MTF for pupil function 105C, illustrating even greater loss ofcontrast due to the phase shifts of segments 107C. Graph 112C showsslices through the 2D MTF also illustrating the loss of contrast. Asampled point spread function (“PSF”) as sampled by a detector is shownin graph 114C. The sampled PSF of graph 114C noticeably broadens (ascompared to the PSF of graph 114B) due to a further reduction in spatialresolution.

MTF curves are plotted in FIG. 6 to further illustrate how opticalsystems are affected by piston error. Each of the four sets of MTFcurves of FIG. 6 are shown both with and without piston error. In graph120, a set of MTFs are shown that correspond to a traditional opticalimaging system 10, FIG. 1; optics 56 employ adaptive optics with pupilfunctions such as shown in FIG. 2-FIG. 4. As shown, the MTFs of graph120 have reduced MTF with increasing spatial frequency and piston error(these MTF curves are the same as the MTF curves of FIG. 3-FIG. 5).

In graph 122, another set of MTF curves are shown that correspond toimaging system 100, FIG. 2, but before processing by decoder 108. Asshown, the MTFs of graph 122 generally exhibit less contrast than theMTFs of graph 120, over most spatial frequencies; however, these MTFsalso have less variation over the range of spatial frequencies due topiston errors. Notice in particular that the MTFs of graph 122 areessentially constant regardless of the piston error. Notice also thatwhile piston error causes zeros in the MTF of graph 120, piston errorscause no zeros in the MTFs of graph 122. As the MTF is a representationof image information, zeros in the MTF are equivalent to a loss of imageinformation. The optical imaging system shown in graph 122 thereforeremoves the information loss caused by the MTF zeros caused as shown ingraph 120.

In graph 124, another set of MTF curves are shown representing MTFcurves after filtering by decoder 108; these MTF curves exhibit nearlythe contrast of the diffraction limited MTFs of graph 120 (in thetraditional imaging system) without piston error. Graph 126 illustratesthis comparison in greater detail, showing that optical imaging system100 provides high contrast over an extended depth of focus whileminimizing the effects of piston error. The filtering provided bydecoder 108 assumed no information about the particular piston error. Ifdecoder 108 has access to the amount of piston error, or access to anestimate of the amount of piston error, even better results arepossible. Decoder 108 can be viewed as an information decoder of theformed imagery. The more information that decoder 108 can access aboutthe formation of the image, the better the decoding process can be.

FIG. 10 shows an exit pupil phase function (equivalent to surfaceheight), and general system parameters, for the optical system describedin FIG. 6 and FIG. 7B. This pupil function has an asymmetric form and ismathematically represented by eighteen of the first twenty-one Zernikepolynomials as shown.

FIGS. 7A and 7B show sampled PSFs for both traditional system 10, FIG. 1and system 100 (employing wavefront coding), FIG. 2, each affected bypiston error due to adaptive optics with pupil functions 105 of FIG.3-FIG. 5. The sampled PSFs of FIG. 7A and FIG. 7B correspond to the MTFsof FIG. 6. In particular, conventional imaging system 10 results in PSFsas shown in FIG. 7A (these are the same PSFs shown above in FIG. 3-FIG.5), which demonstrate a broadening of the PSF (reducing spatialresolution) as each segment of piston error is encountered. Incomparison, optical imaging system 100 with optics 102 (and mask 103)and decoder 108 results in PSFs as shown in FIG. 7B. Again, decoder 108had no information about the amount of piston error. The PSFs of FIG. 7Bdemonstrate little change of resolution as a function of segmentedpiston error. It should therefore be apparent that the foregoingprovides a solution to correct certain aberrations caused by non-idealadaptive optics. Wavefront coding by optics 102 facilitates thiscorrection.

The foregoing paragraph may also hold true for other types of errors,including problems associated with electromechanical orpressure-mechanical actuators that move segments of the adaptive optics.Such errors may be denoted herein as “stuck actuator” errors. Certainother errors are denoted as “quilting errors,” which are caused by aplurality of actuators behind a deformable mirror. Stuck actuator andquilting errors can cause a significant decrease in system MTF, yieldinglow quality final images.

Quilting errors are modeled below using a periodic array of Gaussiandisturbances, each with a half-wave of wavefront error. Stuck actuatorerrors are also modeled using a single Gaussian disturbance, with awavefront error peak value of five waves. With this modeling, FIG. 8shows pupil maps 200 and resulting MTFs 202, illustrating contrastperformance with and without the stuck actuator and quilting errors.Pupil map 200A shows an error-free pupil; its associated MTF has thehighest contrast. Pupil map 200B corresponds to a pupil with quiltingerror; its associated is degraded from the error-free MTF. Pupil map200C corresponds to a pupil with a stuck actuator error; its associatedMTF is further degraded, as shown.

FIG. 9 illustrates how MTF is improved through wavefront coding, such asthrough processing by system 100, FIG. 2. Phase mask 103 is configuredwith an appropriate surface function to modify the wavefront phase andgenerate pupil functions 204A, 204B: pupil function 204A is particularlywell suited to controlling quilting errors; pupil function 204B isparticularly well suited to controlling stuck actuator errors. FIG. 9also shows corresponding MTFs in graphs 206A, 206B: graph 206Aillustrates MTFs with quilting errors; graph 206B illustrates MTFs withstuck actuator errors. In graph 206A, if optics 56 of system 10, FIG. 1,includes adaptive optics with quilting errors, an MTF 208A may result.In graph 206B, if optics 56 of system 10, FIG. 1, includes adaptiveoptics with stuck actuator errors, an MTF 208B may result.

The other MTFs of graphs 206A, 206B result from processing within system100, FIG. 2, when optics 102 include adaptive optics 105 with quiltingand stuck actuator errors, respectively with exit pupils 204A and 204B.In graph 206A, MTF 210A represents MTFs with and without quilting errorsand prior to filtering by decoder 108. In graph 206B, MTFs 210Brepresent MTFs with and without stuck actuator errors and prior tofiltering by decoder 108. In graph 206A, MTFs 212A represent MTFs withand without quilting errors but after filtering by decoder 108. In graph206B, MTFs 212B represent MTFs with and without stuck actuator errorsbut after filtering by decoder 108. MTFs 212A, 212B thus illustrate thatthe resulting MTFs within system 100 (after wavefront coding by mask 103and post processing by decoder 108) are approximately the same as theerror free MTF 214 (corresponding to pupil 200A, FIG. 8), thus providingnear ideal image quality irrespective of quilting and stuck actuatorerrors. Note that MTFs 202, FIG. 8 (also shown in FIG. 9 as 214, 208Aand 208B) vary widely as a function of stuck actuator and quiltingerrors. Once phase is changed at the pupil, by phase mask 103, there areno zeros in the MTF of system 100 and MTFs 210A, 210B are essentiallyconstant with pupil error, demonstrating system invariance to theadaptive optic errors.

Exit pupils 204A and 204B of FIG. 9 result from constant profile pathoptics as set forth in commonly-owned U.S. patent application Ser. No.10/376,924. The specific paths for these optics are four sides of asquare about the optical axis (e.g., axis 109, FIG. 2). Each side ofevery square, or path, has identical form in this example. For exitpupils 204A, 204B, the form of the paths may be described by a secondorder polynomial, each path modulated by a constant that in varies foreach path. The functional form of these ‘across the path’ modulations isthen given by a forth order polynomial. The number of paths in the exitpupil is large, essentially forming a continuous function. For exitpupil 204A, the parameters defining constant profile path parametersare:Along the paths form: C(x)=−5.7667+0.7540x ² ,|x|<1Across the path form: D(y)=0.0561x[−3.3278+35.1536y−34.3165y ²−7.5774y³],0<y<1where the length of each path is considered in normalized unit lengthand the distance from the optical center to the surface edge isconsidered a normalized unit distance. For each path, the same distancefrom the optical center is modulated similarly across the path form. Theparameters for the exit pupil of 204B are:Along the paths form: C(x)=−1.7064+1.2227x ² ,|x|<1Across the path form: D(y)=0.1990x[−6.4896+7.4874y+5.1382y ²−4.9577y³],0<y<1

Although not shown, pupil functions that give similar or improvedresults to pupil functions of 204A and 204B can be formed by thecombination of two or more constant profile path optics.

Other known or unknown aberrations may also be addressed by opticalimaging system 100, FIG. 2. For example, note that the aberration “coma”may have both “focus related” components and other components which arenot focus related. Simulations below show reduction in the effects ofcoma when using system 100 without a priori knowledge of coma. Thefollowing description applies wavefront coding aberration correction interms of optical aberrations up to fifth order and the first thirteenZernike aberration coefficients. The wavefront coding of system 100 mayalso operate to correct and remove other misfocus-like aberrations,including spherical aberration, astigmatism, Petzval or field curvature,axial chromatic aberration, temperature related misfocus, and/orfabrication and assembly related misfocus. Aberrations that are notmisfocus-like are related to coma. The description below offerssolutions to both misfocus-like aberrations and other non focus-relatedaberrations. From this, third and fifth order Seidel wavefrontaberrations are described. Finally, with knowledge of the third andfifth order Seidel aberrations, a partial set of the Zernike wavefrontaberrations can be understood in terms of wavefront coding.

Misfocus-like aberrations have the characteristic that a subset of theaberration may be corrected by movement of the image plane. If allsubsets of the aberration are corrected at the same time, as for exampleby extending the depth of focus, then the effects of the aberration maybe substantially eliminated. Below, we specifically describe themisfocus-like aberrations of spherical aberration, astigmatism, Petzvalcurvature, axial chromatic aberration and temperature relatedaberrations.

FIG. 11 illustrates spherical aberration from an optical element in theform of a lens 230. The spherical aberration causes different radialzones 232 (232A, 232B) to focus at different positions along a range 236of an optical axis 234. Zones 232 result in a change of focus alongrange 236 due to misfocus associated with the zones. Sphericalaberration is a misfocus-like aberration since each zone 232 of lens 230can theoretically be brought into correct focus by movement of the imageplane 231. All zones 232 can be in correct focus at the same time if thedepth of focus is extended to cover range 236.

FIG. 12 illustrates astigmatism from an optical element in the form of alens 240. Astigmatism causes orthogonal axes of lens 240 to come to afocus at different positions 242 along the optical axis 244. Thisaberration is again a misfocus-like aberration since each axis cantheoretically be brought into proper focus through movement of the imageplane (along range 246). If the depth of focus is large enough to coverrange 246, so that light from both axes are in proper focus, then theeffects of astigmatism are substantially removed.

FIG. 13 illustrates Petzval curvature from an optical element in theform of a lens 250. Petzval curvature images a planar object to a curvedimage 252. Object points at different radial distances from the opticalaxis 254 are therefore essentially imaged with different misfocusvalues, i.e., over a range 256. Petzval curvature is a misfocus-likeaberration since each point of a planar object can theoretically bebrought into correct focus by movement of the image plane over range256. If the depth of focus is large enough to cover range 256, theeffects of Petzval curvature can be substantially eliminated.

FIG. 14 illustrates axial chromatic aberration from an optical elementin the form of a lens 260. Axial chromatic aberration causes a bestfocus position 262 to be a function of wavelength or color of theillumination; for example 262A is the best focus position for red light,while 262B is the best focus position for blue light. The focus spreadover the range of wavelengths results in a defocus range 264. Axialchromatic aberration is a misfocus-like aberration since movement of theimage plane can theoretically bring the image formed at each color intoproper focus. If the depth of focus is extended so that the images atall colors are in focus over range 264, then the effects of axialchromatic aberration can be substantially eliminated.

FIG. 15 illustrates temperature related aberrations in association withan optical element in the form of a lens 270. Temperature-relatedaberrations are due to changes in physical lengths, distances, anddiameters as well as to changes in index of refraction of the associatedoptical materials. Such changes may, for example, occur due toenvironmental temperature change that expands or contracts lens 270and/or associated opto-mechanical structure. In one example, lens 270expands or contracts due to such temperature change, such as illustratedby outline 272. In another example, the mounting structure expands orcontracts due to such temperature change, as shown by expansion line274. In still another example, an index of refraction of lens 270 maychange. Certain optical imaging systems may therefore model thermalvariations as a change in best focus position as a function oftemperature, e.g., over defocus range 276 (dependent on temperature).For certain other optical systems, other aberrations such as sphericalaberration and astigmatism are also introduced by changes intemperature. Temperature-related aberrations may therefore be consideredmisfocus-like aberrations since theoretical movement of the image planeas a function of temperature can reduce the effects of temperaturechange. If the depth of focus is large enough to cover range 276 (and,if desired, the other aberrations such as spherical aberration andastigmatism), then the effects of temperature can be substantiallyeliminated.

FIG. 16 illustrates coma in association with an optical element in theform of a lens 280. Coma is an off-axis aberration where different zonesof the lens image with different magnifications. The effects of comaincrease linearly with the distance of the object point from the opticalaxis, causing blurring 282 along the field axis 283 of the image plane285.

Coma is a special aberration different from the misfocus aberration,such as field curvature and chromatic aberration. If decoder 108 employsonly a linear filter, wavefront coding by system 100 may not completelyeliminate the effects of coma; thus some characteristic imaging effectsof coma can be unchanged by the addition of wavefront coding. FIG. 17provides graphical examples of PSFs caused by coma within both imagingsystem 10 and imaging system 100, each as a function of misfocus. Inparticular, FIG. 17 shows a comparison of PSFs for one wave of coma andmisfocus within system 10 (part 290A) and within imaging system 100(part 290B); FIG. 17 also shows a comparison of PSFs for two waves ofcoma and misfocus within system 10 (part 292A) and within imaging system100 (part 292B). The amount of misfocus within FIG. 17 varies from zeroto one wavelength, left to right. The wavefront coded optics (i.e., thephase form of phase mask 103) and signal processing within decoder 108are not made with a priori knowledge of the amount of coma. Notice,however, that the effects of a small amount of coma are reduced but noteliminated within parts 290B, 292B, indicating improvement over imagingsystem 10.

FIG. 17A describes the phase function at the exit pupil of the opticalsystem (mask 103) that formed the wavefront coded images 290B and 292B(shown after decoding 108) of FIG. 17. This phase form is represented inpolar coordinates with five terms. The phase form is the sum of the fiveterms with their corresponding weights. The peak to valley phasedeviation for this phase function is about one wavelength.

To achieve imaging as in parts 290B, 292B, phase mask 103 may, forexample, employ a non-separable aspheric phase at the exit pupil ofsystem 100; signal processing by decoder 108 may then perform a reverseconvolution on the image data to generate the PSFs of parts 290B, 292B.The non-separable phase of phase mask 103 means that decoder 108utilizes non-separable 2D signal processing. The optical resolution andthe resolution of detector 106 are assumed to be matched in order tocritically sample the object. Notice that the blur size is accentuatedwith misfocus within parts 290A, 292A of system 10. In contrast, opticalimaging system 100 generates a slightly reduced blur at zero misfocusand then changes very little with misfocus (in parts 290B, 292B)compared to the changes in optical imaging system 10. While the effectsof coma have been reduced with wavefront coding, the reduction ofmisfocus effects is essentially unaffected by the addition of coma.

To more fully understand the special nature of the aberrations trefoiland coma, as described in Table 1, consider the graphs of FIG. 17B.Graph 170A shows MTFs resulting from misfocus effects of adiffraction-limited system with misfocus varying from 0, ½, to 1 wave.Over this range the MTF changes drastically and even has an MTF zero for1 wave of misfocus. Graph 170B shows the MTFs for a diffraction-limitedsystem that additionally has one wavelength of trefoil aberration overthe same range of misfocus. The form of this trefoil aberration islisted as #9 in Table 1. Notice that trefoil causes a drop in the MTF atall misfocus values, but the change in misfocus is much less than thatshown in graph 170A. Graph 170C shows the MTFs for a diffraction-limitedsystem that additionally has three wavelengths of coma. The form of thiscoma aberration is listed as #6 and #7 in Table 1, with both aberrationsbeing added in the same proportion. Notice that coma also causes a dropin the MTF at all misfocus values, but the change in misfocus is muchless than that shown in graph 170A. Notice also that the MTFs in Graphs170B and 170C have no zeros in the MTF shown. The MTFs shown in graph122 (FIG. 6) show the same change of MTF with aberration. Thus, theaddition of trefoil and coma, for at least some proportion of theseaberrations, act to make the overall system insensitive to effects ofmisfocus. It is then possible to use trefoil and coma as part of Mask103 in system 100.

Although trefoil and coma can be used solely as phase functions for Mask103 in system 100, other combinations that can give improved imagingperformance are possible. Consider the MTF graphs of FIG. 17C. Graph171A shows the MTFs as a function of misfocus for system 100 with thewavefront phase function of FIG. 17A. The long lower MTFs represent theMTFs before linear filtering, the MTFs that are only plotted out tospatial frequency value of 18 are the MTFs after linear filtering. TheMTFs of 171A are high for all values of misfocus, but show a smalleramount of change with misfocus compared to system 10 of graph 170A.Graph 171B shows the MTFs as a function of misfocus for the system thathas the phase function of FIG. 17A plus the trefoil aberration of graph170B. Graph 171C shows the MTFs as a function of misfocus for the systemthat has the phase function of FIG. 17A plus the coma aberration ofgraph 170C. Notice that both the addition of trefoil and coma to thephase of FIG. 17A show increased insensitivity of misfocus effects ingraphs 171B and 171C. Notice also that the addition of trefoil and comaacted to slightly reduce the MTFs from those of graph 171A, when onlythe phase of FIG. 17A is used. Decoder 108 of system 100 could bedifferent for three versions of Mask 103 composed of three differentphase functions of FIG. 17C. The MTFs show that these versions of Mask103 can act to preserve object information by removing MTF zeros, butthe change in MTF for the three versions of Mask 103 may dictate changesin the operation of decoder 108. Decoder 108 could act to estimate thesechanges directly from the images of detector 106 or could be informed ofthe changes by an external source and change accordingly.

If optics 102 of system 100 contains trefoil and coma aberrations,system performance can often be improved by the addition of specializedaberrations. This is shown by PSFs after filtering for a variety of PSFsas a function of misfocus, in FIG. 17D. The misfocus values are 0, ½,and 1 wavelength as in FIGS. 17, 17B and 17C. Graph 172A of FIG. 17Dshows PSFs after linear filtering with Mask 103 being composed of thephase from Graph 171B. Decoder 108 was configured to perform linearfiltering that resulted in a high quality PSF at zero misfocus. Thissame filter was applied to the other misfocus PSFs as well. Notice thatthe PSFs of graph 172A are compact with little change as a function ofmisfocus. Graph 172B shows the PSFs resulting when mask 103 onlycontains the trefoil aberration from graph 170B. Decoder 108 is againchosen to produce a high quality PSF with no misfocus through linearfiltering. Notice that the PSFs of graph 172A are more compact as afunction of misfocus compared to the PSFs of graph 172B. The addition ofthe phase function of FIG. 17A acts to smooth the phase response of theoptical system (not shown) when trefoil aberration is present in optics102. The same is true when coma is present in system 102 as shown bygraphs 172C and 172D. The PSFs after filtering when optics 102 containsonly coma (the amount and form of coma as in graph 170C) are not ascompact when optics 102 also contains the phase function of FIG. 17A.

Therefore, the special trefoil and coma aberrations can be used alone inoptics 102 of system 100, but PSF and/or MTF can often be improved bythe addition of other aberrations in optics 102.

With an overview of misfocus-like aberrations, a relationship may beformed between the third and fifth order Seidel wavefront aberrations.The Seidel aberrations allow the decomposition of wavefront aberrationsinto component aberrations that have physical significance to primaryoptical errors. While the third and fifth order Seidel aberrations arenot orthogonal, they do allow considerable insight into wavefrontaberrations for many types of imaging systems. Below, we describe thethird and fifth order Seidel aberrations and their relationship tomisfocus-like aberrations. Table 2 shows Third Order Seidel Aberrations.Table 3 shows Fifth Order Seidel Aberrations. TABLE 2 AberrationMathematical Name Coefficient Form Relationship to Misfocus-likeAberrations Piston W₀₀₀ 1 If piston is constant over the pupil, then noeffect on the image Defocus W₀₂₀ p² Original misfocus-like aberration.Tilt W₁₁₁ H p cos(θ) If tilt is constant over the pupil then the entireimage is shifted. No other effect on the image or wavefront coding.Field Dependent Phase W₂₀₀ H² Has no effect on the image. SphericalAberration W₀₄₀ H p⁴ Misfocus-like aberration Coma (third order) W₁₃₁ Hp³ cos(θ) Special aberration Astigmatism W₂₂₂ H² p² cos(θ)²Misfocus-like aberration Field Curvature W₂₂₀ H² p² Misfocus-likeaberration Distortion W₃₁₁ H³ p cos(θ) Has no effect on wavefrontcoding. Field Dependent Phase W₄₀₀ H⁴ Has no effect on the image.H represents the height of the image point. (p,θ) represent pupil polarcoordinate variables.

TABLE 3 Aberration Mathematical Name Coefficient Form Relationship toMisfocus-like Aberrations Fifth order Spherical W₀₆₀ p⁶ Misfocus-likeaberration Aberration Fifth order Coma W₁₅₁ H p⁵ cos(θ) Specialaberration Fifth order Astigmatism W₄₂₂ H⁴ p² cos(θ)² Misfocus-likeaberration Fifth Order Field W₄₂₀ H⁴ p² Misfocus-like aberrationCurvature Fifth Order Distortion W₅₁₁ H⁵ p cos(θ) Has no effect onWavefront Coding Sagittal Oblique Spherical W₂₄₀ H² p⁴ Misfocus-likeaberration Aberration Tangential Oblique W₂₄₂ H² p⁴ cos(θ)²Misfocus-like aberration Spherical Aberration Cubic (Elliptical) ComaW₃₃₁ H³ p³ cos(θ) Special aberration Line (Elliptical) Coma W₃₃₃ H³ p³cos(θ)³ Special aberration Field Dependent Phase W₆₀₀ H⁶ Has no effecton the image.H represents the height of the image point. (p,θ) represent pupil polarcoordinate variables.

There are four types of coma in the third and fifth order Seidelaberrations that are special aberrations that act in some proportions todecrease the sensitivity to misfocus effects, as shown above. Linearphase aberrations such as tilt and distortion are not directly correctedby wavefront coding; however, linear phase aberrations typically do notcontribute to a loss of resolution, as other aberrations can. Certainaberrations, such as piston and constant phase, have no noticeableeffect on the image when constant over the entire exit pupil. If thepiston and phase terms vary over the exit pupil, as is found insegmented adaptive optics (described above), then the resultingwavefront aberration may be decomposed into the component aberrationsand analyzed. From the Seidel aberrations, the relationship between theterms of the Zernike aberration polynomials and the misfocus-aberrationsmay be found.

The Zernike aberrations are an orthogonal polynomial decomposition overa circular area. Their orthogonal nature makes the Zernikes a usefultool for many forms of analysis and optimization. Table 4 shows thefirst 13 Zernike polynomial terms, and describes their relationship tothe Seidel aberrations and misfocus-like aberrations. TABLE 4Relationship to Term # Mathematical Form Relationship to SeidelAberrations Misfocus-like Aberrations 1 1 Piston Does not effect image 2p cos(θ) Tilt Wavefront coding not affected by tilt. 3 p sin(θ) Rotatedversion of #2 Wavefront coding not affected by tilt. 4 2 p² − 1 Misfocus& Piston Misfocus-like aberration 5 p² cos(2θ) Astigmatism & MisfocusMisfocus-like aberration 6 p² sin(θ) Rotated version of #5 Misfocus-likeaberration 7 (3p² − 2) p cos(θ) Coma & tilt Special aberration 8 (3p² −2) p sin(θ) Rotated version of #7 Special aberration 9 6p⁴ − 6p² + 1Spherical Aberration & Misfocus & Piston Misfocus-like aberration 10 p³cos(3θ) Special aberration 11 p³ sin(3θ) Special aberration 12 (4p² − 3)p² cos(2θ) Tangential Oblique Spherical Aberration Misfocus-likeaberration & Astigmatism & Misfocus 13 (4p² − 3) p² sin(2θ) Rotatedversion of #12 Misfocus-like aberration(p,θ) represent pupil polar coordinate variables.

The Zernike terms #7 and #8 are related to coma and thus are specialwavefront coding terms that can be used to control the effects ofmisfocus as described above. The Zernike terms #10 and #11 are alsospecial in the same manner. One particular wavefront coding surface isgiven in polar coordinates as p³ cos (3θ). This term in rectangularcoordinates is related to the form [X³+Y³−3(XY²+X²Y)]. The cubic terms[X³+Y³] can be used to form a rectangularly separable wavefront codingsurface (on phase mask 103). Higher order Zernike terms are composed ofseventh and higher order Seidel aberrations and are not shown.

With Zernike wavefront analysis, the root-mean-square (RMS) wavefronterror is calculated as the RMS value of the weights of each Zernikepolynomial term. Wavefront coding by system 100 may allow an effectiveRMS wavefront error where the weights of the misfocus-like aberrationsin the Zernike expansion are considered zero since the effect can becontrolled with wavefront coding. In one embodiment, the effective RMSwavefront error with wavefront coding may consider the weights on theZernike polynomial terms 4, 5, 6, 9, 12 and 13 to be equal to zero. Inother configurations, where decoder 108 can be dynamic, the Zernikepolynomial terms 7, 8, 10 and 11 can also be considered to be equal tozero.

Wavefront coding has other advantages in optical imaging systems. Forexample, when used within thermal sighting systems, such as withinreconnaissance aircraft, the inclusion of wavefront coded optics candiffuse or diminish system responsiveness to unwanted radiation, such asfrom a targeting laser. By way of example, FIG. 18 shows one thermalsighting system 300, which includes an optical imaging system 302 withone or more optics 304, optics 306 (e.g., employing phase mask 103, FIG.2), and a detector 308 (detector 308 may be a human eye, or otherdetector such as a CCD or a thermal detector (e.g., an InSb array)).Optics 304 and wavefront coded optics 306 need not be separate physicalelements in practice. Sighting system 300 can, in general, operate overany band on the electromagnetic spectrum. Optical imaging system 302serves to diffuse incoming radiation 310 so as to reduce possiblenegative effects on detector 308. In particular, although optics 304 maybe such that radiation 310 is focused on detector 308, optics 306operate to disperse radiation 310 at detector 308. Post processingwithin a decoder 312 (e.g., decoder 108, FIG. 2) then serves to recreatean image of optics 306 in a clear manner. Post processing at decoder 312may occur within a human brain if detector 308 is a human eye.

FIG. 19 shows one illustrative optical imaging system 302A, including aoptics 304A and wavefront coding element 306A (which in this example isa surface of optics 304A). The system parameters of system 302A areshown in table 320. Also shown are two exemplary phase forms suitablefor use in forming wavefront coding element 306A: phase form 322 is aconstant path profile surface; phase form 324 is an odd asphericsurface. Incident radiation 310A is dispersed by optics 304A/306A,reflected off of detector 308A, and dispersed out of system 302A, asshown by rays 309.

Addressing the problem of unwanted reflections in an optical imagingsystem by conventional means typically introduces unwanted aberrationsinto the system. FIG. 20A shows a prior approach that utilizes lowreflectivity of an optical imaging system 350. Incident optical waves orrays 352 are focused by optics 354 onto a focal plane array detector 356in system 350. Detector 356 back scatters at least some portion of rays352, for example when detector 356 has some characteristics of aLambertian reflector. A prism 358 may be inserted into system 350 neardetector 356 to bend rays 352 so that they are reflected back throughsystem 350, such that the reflected rays are blocked by an aperture stop360 of system 350.

A disadvantage of the methods of FIG. 20A is that prism 358 introducesaberrations into system 350. A ray intercept map related to system 350is shown in FIG. 20B and demonstrates the image aberrations introducedby placing prism 358 into system 350. Without prism 358, the rayintercept map would consist of highly concentrated points much closertogether than those of FIG. 20B. While aberrations introduced by prism358 may be reduced by positioning prism 358 as close to detector 356 aspossible (i.e., at the focal plane of system 350), the aberrations arenot fully corrected due to the spatial variation, or separation, of rays352 from different field points.

FIG. 21A shows an optical imaging system 400 in which a tilt surface 408is introduced in optics 404, or in other optics (not shown) adjacent tooptics 404, at an aperture stop 410 of system 400. Optical waves or rays402 incident on optics 404 are focused onto a focal plane array detector406 in system 400. Reflections of rays 402 back scattered from detector406 are deviated by tilt surface 408 such that the reflected rays 402are substantially blocked by aperture 412 surrounding optics 404, andbefore such reflections propagate into object space 414. Tilt surface408 is tilted away from a plane perpendicular to the path of travel ofrays 402. Optical imaging system 400 therefore does not suffer from thesame shortcomings as system 350 with prism 358, since the tiltintroduced by tilt surface 408 at aperture stop 410 ensures that thereis no spatial variation between reflections of rays 402 coming fromdifferent locations in object space 414. Since all of the fields see thesame part of tilt surface 408, additional aberrations due to the tiltcan be corrected by processing an image detected by detector 406 withpost processor 416 (e.g., decoder 108, FIG. 2). Post processing 416 hasa priori information about the degree of tilt of tilt surface 408. FIG.21B shows a ray intercept map of rays 402 in system 400. Notice that theenergy of rays 402 is concentrated near the on-axis position, showingthe desired co-focusing of rays. In an equivalent system detector 406may be tilted, instead of a tilt being added to surface 408.

Further reduction of reflectivity in optical imaging system 400 may berealized by configuring optics 404 with wavefront coding (e.g., such asoptical imaging systems employing mask 103, FIG. 2, or wavefront codingelement 306A, FIG. 19). Even with the addition of tilt surface 408,system 400 does not block all reflected rays 402 from travelingbackwards through optics 404 to object space 414. Optics 404 uniformlyblur rays 402 such that the energy of rays 402 entering system 400 arespread out prior to reaching detector 406. Because ray energy is lessconcentrated, there are fewer Lambertian-like reflections from detector406. Wavefront coded optics 404 may further be selected with a phasefunction such that the few rays 402 reaching optics 404 are blurred toreduce reflections.

FIG. 22 shows typical intensity profiles for the reflected ray energy ata far field (e.g., 500 meters) with a conventional optical imagingsystem and with wavefront coding by system 302A, FIG. 19. Plot 410corresponds to phase form 322; plot 412 corresponds to phase form 324.The intensity of reflected energy 309, or irradiance, is much lower whenwavefront coded optics 306A are used in optical imaging system at anon-axis position as compared to a conventional optical imaging system(e.g., system 10, FIG. 1). Even at off-axis positions, the irradiance ofreflected energy 309 is lower.

A further example of the anti-reflection performance possible withwavefront coding is shown in FIGS. 22A, 22B, 22C, and 22D. FIG. 22Ashows the integrated reflected power from an example optical imagingsystem (employing wavefront coding) and a traditionaldiffraction-limited system. The simulated system has a working F/# of0.9, 10 micron illumination, and 25 micron square pixels with 100% fillfactor. The integrated reflected power is represented as a function ofangle at the far field of the sensor. The vertical scale is in dB and isgiven as 10*log₁₀(reflected energy). The integration area is ¼ of thewidth of the reflected main lobe of the traditional ordiffraction-limited system. The reflected power is approximately 42 dBless for the optical imaging system as compared to the traditionalsystem, at an angle of zero as shown in FIG. 22A.

The MTFs related to the imaging system 300 (from FIG. 18) that generatedthe graph of FIG. 22A is shown in FIG. 22B. The wavefront coding phasefunction reduces the MTF (and also greatly reduces the reflected on-axisenergy). After processing by decoder 312 the MTF is increased to asystem specified level. For this particular system, the MTF at thedetector cutoff frequency (denoted at the normalized spatial frequencyof 1.0 on FIG. 22B) was 0.4. Notice that the wavefront coded MTF beforefiltering of FIG. 22B also had no zero values within the passband of theimage detector.

The imaging exit pupil (or phase function added to thediffraction-limited system) related to FIG. 22A is shown in FIG. 22C.This exit pupil has about one wave peak-to-valley phase deviation and isconstructed as the sum of two constant profile path phase functions. Asthis exit pupil has 180 degree symmetry, the equivalent exit pupil forthe reflected energy is twice the imaging exit pupil. The mathematicalform of this sum of two constant profile path exit pupils is, for thisexample, defined as:Along the paths form #1: C(x)=0.645−1.95x ²+3.45x ⁴−1.80x ⁶ ,|x|<1Across the path form #1: D(y)=1,0<y<1Along the paths form #2: C(x)=1,|x|<1Across the path form #2: 3.79+2.87x−6.29x ³+2.80x ⁴ ,D(y)=,0<y<1

A mesh view 500 and an image view 510 of a sampled PSF related to theimaging exit pupil of FIG. 22C are shown in FIG. 22D. The sampled PSF isseen to be spatially compact. Decoder 312 acts on the sampled PSF ofFIG. 22D to produce high quality images. Decoder 312 can be specializedto the use of the produced images. If a human is viewing the images,decoder 312 may have no operation, as the human brain can be used toremove the spatial effects. As the imaging MTF from FIG. 22B has nozeros, all object information is contained in the sampled PSF of FIG.22D. If a target detection system is viewing the images then decoder 312can be configured to produce the type of images best suited for theparticular target detection system. In general, the target detectionsystem (or image information system) that acts on the produced imagerymay be jointly optimized with decoder 312 to increase overall systemperformance and decrease total costs.

FIG. 23 schematically shows an imaging system 660 for imaging acousticalwaves 662. System 660 has an encoder 664 for coding a wavefront ofacoustical waves 662 incident thereon from a medium 666. Encoder 664makes an imaged wavefront 665 of acoustical waves 662 substantiallyinvariant to acoustical aberrations caused by medium 666. Acousticalsound imager 668 detects encoded acoustical waves 662 and decoder 670removes effects caused by encoder 664 when coding acoustical waves 662.In this way, system 660 generates acoustical sounds 671 that aresubstantially equivalent to sounds that would be obtained if noaberrations were introduced by medium 666.

System 660 thus operates similarly to imaging system 100, FIG. 2. System660 may also be modeled to provide further refinement of the optimalacoustical imaging properties through medium 666, similar to the abovemodeling for system 100.

The following describes software processing suitable with certainoptical imaging systems (employing wavefront coding) that utilizeextended depth of field and/or passive ranging. Such processing is forexample particularly useful in a number of applications where more thanthe minimum amount of signal processing is available such as inminiature cameras, microscopy, biometric imaging and machine visionsystems. In one example of the prior art, a major problem in automatedmicroscopy is the determination of best focus (or range to the object)and the act of securing best focus before an image is acquired andarchived. This problem is complicated in the prior art since the focusposition over the entire specimen (i.e., the object being imaged) mayvary dramatically, requiring refocusing at each image acquisitionlocation. More particularly, in the prior art, best focus is estimatedby acquiring a series of images over various axial positions (z). Afocus score is determined for each image, and an image with the highestfocus score indicates best focus position (or range), or an interpolatedfocus score is used. This process may repeat for differentmagnifications, e.g., for coarse magnification objectives or finemagnification objectives.

Consider optical imaging system 700 of FIG. 24. System 700 has optics702 and an associated wavefront coding element 704 (which may beintegral with optics 702). Optics 702 and wavefront coding element 704encode and focus a wavefront 708 onto a detector 710; wavefront 708 is aconstant phase front from an imaged object 712. Post processing 714serves to post process data from detector 710 to generate a final infocus image 716.

System 700 may, for example, be an automatic microscopy system, used forslide scanning and high throughput screening. System 700 may also be abiometric imaging system used for access control. System 700 may also bea miniature camera where the object is either “far” or “near”,essentially acting as an electronic macro system with no moving parts.System 700 avoids the problems and repeat procedures of the prior art.As described in more detail below, system 700 may employ softwareprocessing in a variety of forms to generate high-quality images.

More particularly, system 700 processes data from object 712 (or, forexample, the “specimen” if system 700 is a microscope) to automaticallyrefocus and possibly range on a single acquired image. This isaccomplished by characterizing optical system 700, then acquiring asingle image and best focus through software processing within postprocessing 714. In general, the expected range of the object exceeds thedepth of field of the imaging system. However, due to optics 702 ofwavefront coding element 704, the MTFs have no zeros over this broadobject range. When the MTF has no zeros, the underlying objectinformation is preserved by optics 702. Post processing 714 then acts ina manner similar to decoder 312 or decoder 108 to decode the properinformation from the sampled image. This allows determination of theobject range, as well as digital processing best suited to theparticular object location. In greater detail, these steps may, forexample, be performed by:

-   -   (1) Acquiring PSF images (or other samples of system images)        over a wide rangeof focus positions (e.g., for a 20×/0.5        objective, acquiring images +/−20 μm from best focus). The range        of focus positions exceeds the depth of field of the system so        that the PSFs can change appreciably over the range of focus        positions.    -   (2) The incremental steps of (1) can be linear or irregular,        based upon PSF rate of change. Building “specific” digital        filters used in post processing 714 for each focus position or        broad range of object positions; “specific” means that the        filter is optimized from only one PSF image or region of focus        positions.

FIG. 25 illustrates step (1). In FIG. 25, a series of PSF images 800taken over a broad range of focus positions is shown. This range exceedsthe depth of field of the imaging system forming the PSFs; therefore,the PSFs change appreciably over this range of focus positions. A filterdesign engine (FDE) process 802 formulates a corresponding set offilters 804, as shown, to create a filter stack 806, one filter for eachregion of focus positions. The process of step (1) may occur in system700 with or without wavefront coding element 702.

In step (3), the following sub-steps may be made, such as illustrated inFIG. 26:

-   -   (3.1) Acquire a single image 800A at one focus position or        object range. The image may be formed from a general scene.    -   (3.2) Filter image 800A with filter stack 806, to create a stack        of filtered images 808.    -   (3.3) Calculate focus score 810 from of stack of filtered images        808.    -   (3.4) Find best focus score 812 (or estimate of object        range/best filter for the given image). In general, the        particular filter from the filter stack 806 that “best” removes        the blur from the sampled image 800A also describes an estimate        of the misfocus amount or range to the object.    -   (3.5) Use the best focused image (e.g. archive or analyze) or        use one additional step 3.6.    -   (3.6) Set a focus position (based on knowledge of particular        amount of focus position from best focus score 812) and retake        an image at best focus. A phase mask 103 can be specifically        used for ranging purposes. This ranging may occur with coarse        focus functions using a low power lens. Fine focus imaging may        occur with higher magnification and numerical apertures, thus        permitting use of a less pronounced phase mask 103 (or no phase        mask 103). Since many optical systems have asymmetrical defocus        (i.e., the response of an ideal imaging system to + and −        misfocus is the same), ranging may be performed on the sampled        images without wavefront coding or on logarithmic, rotationally        symmetric pupils. One advantage of using phase mask 103 (versus        no mask) is that the PSF with wavefront coding has no MTF zeros,        providing better information capture at small misfocus values.

FIG. 27 illustrates focus score through use of an asymmetric orsymmetric defocusing phase mask 103. In particular, FIG. 27 shows threegraphs 850A, 850B and 850C illustrating hypothetical focus score versusz for symmetric defocusing (850A), asymmetric defocusing (850B), and apseudo-symmetric defocusing (850C).

Software focusing as described in connection with FIG. 24-FIG. 27 mayprovide certain advantages. For example, system 700 may be a fixed focusminiature camera that exhibits auto-focus and/or macro-focusfunctionality. In another example, system 700 is a cell phone camerathat has a fixed focus which makes good quality portraits and close-upimages of objects 712 (e.g., business cards). Moreover, filter stack 806may be determined a priori over a range of expected object distances. Inone variation, manual adjustment may also be used to determine bestfocus, as an alternative to use of the focus score. The stack of filters(i.e., filter stack 806) need not be linear filters. They could be anycombination of digital processing operations that act on the sampledimage to produce a high quality image suitable for the user (human ormachine) of the image.

It should be clear to those skilled in the art that system 700 may haveother features. For example, it may optically zoom to greater range offocus and magnifications by moving zoom lenses, where focus score isdetermined by a series of filters taken at different zoom positions, toselect best focus. Accordingly, a user may adjust the filters at aparticular zoom to select the preferred or desired object distance. Or,focus score may be used with a matrix of filters, first by zoom, then byobject distance (e.g. rows=zoom, columns=subject distance).

Therefore, system 700 may have further application as asoftware-athermalized optical imaging system, such as within telescopesand IR imagers. It may alternatively serve within machine vision systemsusing fixed focus lenses 702. It may also serve as an endoscope withhardware zoom features but with software optimized filter selection.

In many types of imaging systems 10 of FIG. 1, image processing 60 isspecific, task-based image processing. Such task-based processing isoften used to determine image information. This information may include,for example, the spatial location of objects, lines, edges and/orpoints; the presence or absence of bars and/or squares, and generalimage-based statistical quantities, the latter two types correspondingto task-based image systems such as bar code scanners and biometricrecognition systems, respectively. Structured light imaging systems,where image information is coded onto the spatial location of objectssuch as bars in the image, is another example of a task-based imagingsystem used to determine image information.

As described in more detail below, a version of task-based processing(employing wavefront coding) may also occur with system 100, FIG. 2,with additional image processing 60 after decoder 108. Such a system is,for example, used to produce the task-based image information over alarger depth of field (or depth of focus) as compared to system 10, FIG.1, which under like conditions produces aberrated images due to poorlyperforming lenses over a broader temperature range, etc.

Since specific optical imaging systems are used to determine imageinformation, and do not produce images for human viewing, in oneembodiment, decoder 108 is not included within system 100. Moreparticularly, decoder 108 may have the characteristic that it does notcreate or remove needed image information. The image information isinstead transformed into a form more suitable for human viewing or otherspecialized processing by decoder 108. The amount of image information,or image “entropy,” may also be unchanged by decoder 108. Since theamount of image information may be unchanged by decoder 108, imageprocessing 60 after decoder 108 may be essentially insensitive toprocessing by decoder 108. In other words, certain performance aspectsof task-based wavefront coding optical and digital imaging systems canbe unaffected by either the presence or absence of decoder 108. In suchcases, decoder 108 may be removed from system 100 of FIG. 2 with noadverse effects. That is, the stored electrical representation of theimage output by certain task-based processing does not contain effectsof wavefront coding that would otherwise require explicit processing toremove.

Consider, for example, the task-based imaging system 100A of FIG. 28A.System 100A is illustratively functionally equivalent to system 10, FIG.1, except for specialized task-based processing by task-based imageprocessing 160. System 100A operates to image object 50A to detector 158and accurately estimate the spatial center of object 50A. That is, thestored electrical representation of the object output by system 100A isthe spatial center of object 50A. On (x,y) coordinates known to system100A, the spatial center of object 50A is (x₀,y₀). The output 163 oftask-based image imaging system 100A is an estimate of (x₀,y₀), denotedas (x₀′,y₀′). The actual image formed and seen by a human on detector158 is not important for this example. In this system, and in othersystems, only an estimate of spatial location (x₀,y₀) is desired. Thistask is therefore an example of a task-based system forming images inorder to estimate image information.

Wavefront coding within system 100A is, for example, useful if thedistance between object 50A and system 100A is unknown or varies, i.e.,such that the image formed on detector 158 does not contain sufficientaccuracy to object 50A and estimates (x₀,y₀) are not readily made bytask-based image processing 160. Wavefront coding can also be used toreduce the complexity and cost of imaging optics 156 by removingnegative effects of aberrations due to optics, mechanics, alignment,thermal changes and/or aliasing, each of which can have on the accuracyof the estimate of (x₀,y₀).

System 2000 of FIG. 28B is a version of system 100A (employing wavefrontcoding), FIG. 28A. System 2000 has optics 2002, decoder 2004, andtask-based image processing 2006. Optics 2002 contain image forminglenses 2008 and wavefront coding aspheric optics 2010. Lenses 2008 andoptics 2010 create an image of object 1998 at a detector 2012. Lenses2008 and optics 2010 may be combined so that the total number of opticalelements is one or more. Task-based image processing 2006 can, but doesnot have to, be the same as processing 160 of FIG. 28A.

Now consider one task-based image processing performed by processing2006 in order to generate an estimate 2016 of a center (x₀,y₀) of object1998. The spatial information about the object center can be describedin the spatial domain by a centroid of object 1998. Calculation of theimage centroid can be used to estimate (x₀,y₀). The spatial informationabout the object center can also be described by a linear phase term ofa Fourier transform of the image formed by optics 2002 at detector 2012.As known to those skilled in the art, spatial location is represented inthe frequency domain through the slope of a linear phase component of acomplex Fourier transform of the image. Calculation of the slope of thelinear phase component in the frequency domain yields an estimate of(x₀,y₀). Due to the nature of these frequency domain calculations, theestimate of (x₀,y₀) may be insensitive to the presence or absence ofdecoder 2004.

For example, assume that a spatially centered version of object 1998 isrepresented mathematically (in one dimension for clarity ofillustration) as o(x) with spatial Fourier transform O(u). Assume forthis example that this spatially centered version has no linear phasecomponent in O(u). Also assume that the spatial blurring function ofwavefront coded optics 2002 is given by h(x), with spatial Fouriertransform H(u). Assume also that decoder 2004 acts to minimize thespatial blur h(x) through convolution with a spatial kernel f(x) (with aspatial Fourier transform F(u)). Then, the image of object 1998 withcenter (x₀,y₀) measured at detector 2012 can be approximated as:sampled_image=o(x−x ₀)*h(x)where ‘*’ denotes spatial convolution. In the spatial frequency domain,this sampled image can be represented as:sampled_image′={O(u)exp(j u x ₀)}×H(u)where u is the spatial frequency variable, j is the square root of −1,and ‘x’ denotes point-by-point multiplication. Notice that spatiallocation x₀ is now part of a linear phase term, (u x₀). Any linear phaseterm of H(u), if H(u)=H(u)′exp(j u z₀), can be considered as a knownspatial bias of amount z₀. The sampled image after decoder 2004 in thespatial domain can be approximated as:sampled_image_after_filtering=o(x−x ₀)*h(x)*f(x)With the sampled spatial image after filtering (by decoder 2004), acentroid calculation (within processing 2006) can be used to estimatethe spatial center x₀, since the combination of the system blur h(x) andfilter f(x) are combined to essentially yield, for this example,h(x)*f(x)≈delta(x), where delta(x) is 1 if x=0, and equal to 0otherwise. If the centroid calculation is performed before applyingfilter f(x) (i.e., before filtering by decoder 2004), the spatialblurring by h(x) could yield inaccurate estimates of the spatial centerx₀. Since processing 2006 is a centroid calculation, it is not alonesufficient to remove effects of wavefront coding (of element 2010)without decoder 2004.

The equivalent filtered image after decoder 2004 can be approximated byapplying filter F(u) in the frequency domain. This results in:sampled_image_after_filtering′={O(u)exp(j u x ₀)}×H(u)×F(u)If the filter F(u) has a linear phase term, such that F(u)=F(u)′exp(j uz₁), then the filter adds an additional known bias term z₁ to the linearphase component of the sampled image after filtering. But since themagnitude of F(u) is typically greater than zero, applying filter F(u)does not help or hurt the calculation of the linear phase amount x₀ inthe frequency domain. With or without filter F(u), the process ofestimating x₀ is the same: 1) calculate the spatial Fourier transform ofthe signal, 2) separate the complex phase and magnitude, 3) calculatethe linear phase component, and 4) subtract any system bias. Calculationof the linear phase component can be performed through a least squarestechnique by fitting the phase to a straight line (for 1D images) or aplane (for 2D images). Filter F(u), if |F(u)|>0, alters the calculationof the spatial shift through the addition of a known spatial locationbias z₁, which may be subtracted during calculation of the estimate ofx₀. Therefore, task-based processing 2006 is insensitive to the presenceor absence of decoder 2004 as to determining estimate 2016. Decoder 2004is not required to achieve the benefits of wavefront coding in system2000 compared to that of system 100A without wavefront coding. Morespecifically, the frequency domain processing within processing 2006outputs a stored electrical representation of object 1998 that isinsensitive to effects of wavefront coding that would otherwise requireexplicit processing to remove. FIG. 28C shows system 2000A withtask-based image processing 2006A and without decoder 2004 of FIG. 28B.Optics 2002A contain image forming lenses 2008A and wavefront codingaspheric optics 2010A, which create an image of object 1998A at adetector 2012A. System 2000A is functionally equivalent to system 2000(like numbers providing like functionality), absent decoder 2004;decoder 2004 being absent due to specialized processing 2006A, providinglike output 2016A

The reason that decoder 2004 is not required in system 2000A is thattask-based imaging processing 2006, 2006A is used to determineinformation from the formed images at detector 2012, 2012A,respectively. Such information, also often called entropy, is amathematical term often used in communication systems to describe theamount of bits needed to transmit or describe a signal, e.g., a voicesignal, a radar signal, an image, etc. In image processing, suchinformation is often related to the amount of randomness orun-anticipated aspects of an image. If an image is completely knownbefore being viewed, then this image brings little information to theviewer. If the image is unknown before viewing, the image may bring asignificant amount of information to the viewer depending on the imagingsystem used and the object being imaged. In general, the object containsthe information and the imaging system transfers this information. Sinceimaging systems cannot transfer spatial information perfectly, theinformation contained in the image is typically less than that of theobject. If the imaging system misfocuses, then the MTF of the imagingsystem can have regions of zero power and the system can transfer littleinformation from the object. After sampling by the detector, the amountof image information can only be kept constant or destroyed (reduced)with digital processing (e.g., by decoder 2004 and processing 2006).Digital processing cannot create information previously lost within animaging system; it is only used to change the form of the imageinformation. This concept is termed the Data Processing Inequality (see,e.g., Elements of Information Theory, Cover and Thomas, John Wiley &Sons, Inc, 1991). In one example, a human can have difficulty viewingand understanding a modified image where each spatial frequencycomponent of the image has been deterministically modified with anon-zero phase, even though the amount of information can technically beexactly the same as the un-modified image. In contrast, image processing2006, 2006A of a task-based system can be designed and used such thatdeterministic modifications of the spatial frequency components havelittle to no effect on performance of the task.

Another example of a wavefront coded task-based imaging system is abiometric recognition system, or more specifically, an iris recognitionsystem 3000 of FIG. 29. System 3000 with task-based processing 3600(specialized for iris recognition) images the iris of eye 3001. System3000 is functionally equivalent to system 2000A of FIG. 28C with theexception that task-based processing 3600 is specialized for irisrecognition. Optics 3002 include image forming optics 3003 and wavefrontcoding optics 3004. Both optics 3003 and 3004 can be combined onto thesame element such that optics 3002 contains a minimum of one opticalelement. Detector 3006 detects electromagnetic radiation 3005 imaged byoptics 3002. Task-based processing 3600 produces a sequence of bits, oran iris feature code 3601, related to the complex phase of the image ofiris 3001 convolved with a complex function. In one embodiment, thiscomplex function is a 2D Gabor bandpass function as a function ofspatial location and scale. An image of iris 3001 directly useful to ahuman is not output by task-based processing 3600. Iris feature codes(i.e. iris feature code 3601) take many forms, but all act to arrangefeatures in the iris such that differences in the eye of every person,and even between eyes of the same person, can be determined and suchthat a specific iris can be recognized and others can be rejected.Task-based processing 3600 may therefore act to code the biometricinformation of the iris image into a form practical for recognitiontasks. See U.S. Pat. No. 5,291,560 (March 1994); Biometric PersonalIdentification System Based on Iris Analysis; and “Demodulation byComplex-Valued Wavelets for Stochastic Pattern Recognition”, by JohnDaugman, International Journal of Wavelets, Multiresolution andInformation Processing, Vol. 1, No. 1, (2003) pg 1-17, each of which isincorporated herein by reference, for more information about irisfeature codes and iris recognition.

Task-based processing used to generate iris feature codes can be largelyindependent of the presence or absence of wavefront decoder 2004 of FIG.28B. In some cases, the absence of decoder 2004 is preferred in irisrecognition because of noise amplification effects that can beintroduced by decoder 2004. Consider an example iris that can bedescribed in the spatial domain as:Iris=I(x)where again 1D representations are used for ease of illustration.Extension to 2D representations are apparent to those skilled in the artof signal processing. Consider an amount of spatial blurring of theimaging optics as the function h(x). Then the sampled iris image can bedescribed as:Iris_image=I(x)*h(x)+n(x)where again ‘*’ denotes spatial convolution. The term n(x) is a noiseterm that is present in all real imaging systems. This noise can be dueto detector additive noise, pixel non-linearity and non-uniformity,image jitter, iris movement, etc. For this representation, both additiveand multiplicative noise is represented as additive noise for ease ofillustration. Let the complex feature code forming function, which couldbe a complex Gabor wavelet, a complex Haar wavelet, and many others, bedenoted as c(x)_(ik) where i and k are indexes related to the particularparameters of the feature code forming function. The feature codeforming function is applied to the iris image to yield the Iris FeatureCode:Iris_Feature_Code_(ik)=Phase[c(x)_(ik) •{I(x)*h(x)+n(x)}]where the process Phase[ ] calculates the complex phase of the IrisImage acted on by particular iris feature code forming function. Thesymbol ‘•’ denotes the general operation performed by the particulariris coding scheme. This could be multiplication, convolution(smoothing), or other operations. Often this complex phase is quantizedinto two bit sequences relating to the possible four quadrants of a unitcircle.

For a given iris, the Iris Feature Code is a statistical quantity with acertain mean, variance and probability density. Consider the differencesin the output of the feature code forming function and the iris imagewith a spatial blurring function from an in-focus diffraction limitedimaging system h(x)_(dl), and the spatial blurring function from a welldesigned wavefront coding system h(x)_(wfc). The well designed wavefrontcoding system has no zeros in its MTF over a broad range ofmisfocus-like aberrations over the spatial passband of the digitaldetector:c(x)_(ik) •{I(x)*h(x)_(dl)}vs. c(x)_(ik) •{I(x)*h(x)_(wfc)};If we assume that the operator • denotes point by point multiplicationthen we can write, in matrix notation:C^(T) _(ik) H _(dl) I vs C^(T) _(ik) H _(wfc) Iwhere C _(ik) is a vector representing the feature code formingfunction, I is a vector representing the iris, and H are convolutionmatrices. The superscript T denotes transpose. The formed iris images(H_(dl) I) and (H_(wfc) I) are different versions of iris featureinformation from the same iris. Since the MTF of the wavefront codingsystem was designed so that the MTF has no zeros, there exists a linearfilter and convolution matrix H_(f) such that:H_(f) H_(wfc) I=H_(dl) Iwhere the convolution of the filter with the sampled wavefront codediris image is essentially the same as an iris image from a diffractionlimited (or any other) iris image in the absence of noise. Knowledge ofthe wavefront coded iris image is sufficient to form the iris image thatwould have been formed by the in-focus diffraction-limited image. So,the features of the iris can be thought of as being reformatted by adeterministic blurring function of the wavefront coding system. Nofeatures of the iris are lost, merely rearranged. If decoder 2004 isused, then the iris information can be explicitly formatted to thatexpected from the diffraction-limited system.

Counting the fraction of bits that differ in two iris images is a commonmetric to measure differences in iris feature codes. The fraction canvary from 0 (no differing bits) to 1 (all bits differ). This metric iscalled the Hamming distance. The expected Hamming distance from twonoise-free iris feature codes of the same iris can be essentially thesame when both iris images are formed with an in-focusdiffraction-limited system, when both iris images are formed from awavefront coded system without decoder 2004, when both iris images areformed with a wavefront coded system where decoder 2004 is used, or whenone image is formed with an in-focus diffraction-limited system and theother is formed with a wavefront coding system with decoder 2004. In thelatter case, decoder 2004 acts to form an equivalent code that measuredby the diffraction-limited system. The expected Hamming distance betweentwo iris feature codes of different irises when both are imaged with anin-focus diffraction-limited image can also be essentially the same aswhen the set of iris images are formed with a wavefront coding systemwithout decoder 2004, or when the set of iris images are formed with awavefront coding system with decoder 2004. The expected Hammingdistances between iris feature codes from the same or different iriswhen imaged with two different imaging systems do not have this idealcharacteristic. The ideal characteristics are present when sets of irisimages are formed with the same type of imaging system. The performanceof the noise-free iris feature codes, in terms of the expected Hammingdistance, can be essentially the same when imaged with thediffraction-limited system or a wavefront coding system with or withoutdecoder 2004. That is, the stored electrical representation of the image(the iris feature code) does not contain effects of wavefront codingthat would otherwise require explicit processing to remove (due tospecialized processing of processing 3600).

If decoder 2004 of FIG. 28B is used before the feature code formingfunction, the image just after the decoder can be represented as:{I(x)*h(x)_(wfc) +n(x)}*f(x)=H _(f) H _(wfc) I+H _(f) nwhere in this case decoder 2004 applies a linear digital filter f(x).Notice that the decoder acts on the term containing the iris and theterm containing the noise. As above, the decoder merely rearranges theform of the iris features, but the noise term after decoder 2004 is nowspatially correlated. If we assume for simplicity that the noise isindependent white Gaussian noise with zero mean and variance σ², afterdecoder 2004, the noise is spatially correlated with correlation givenby:Noise correlation=σ²H_(f)H^(T) _(f)where ‘T’ again denotes transpose. A grammian (H_(f)H^(T)) of thedecoder convolution matrix now forms the noise spatial correlation. Thenoise after the decoder may not be independent and white but may bespatially correlated due to the action of decoder 2004. Without decoder2004, the noise in the iris feature code calculation is uncorrelated andindependent for each spatial position and scale. With decoder 2004, thenoise in the iris feature code calculation may become correlated withspatial position and scale. This noise correlation may act to remove theefficiency of the estimates of the iris feature code, resulting in aloss in information in the feature code, depending on the particulariris feature code. In essence, spatially correlated noise in the irisimages results in the addition of noise-dependent statistical features.The noise features can make the expected Hamming distance between irisfeature codes of iris images of the same iris increase (seem moredifferent) and decrease (seem more similar) the expected Hammingdistance between iris feature codes of iris images of different irises.In one case then, decoder 2004 acts to reformat the noise-free irisfeature codes to be similar to that from the in-focusdiffraction-limited system, but also makes the task of iris recognitionand rejection in the presence of noise more difficult. For this type oftask-based processing, decoder 2004 can be specialized or optional, withsome systems preferring the absence of decoder 2004.

If noise n(x) directly from detector 2012, 2012A is spatiallycorrelated, a form of processing q(x) and H_(q) may be used beforefeature code formation, possibly in decoder 2004, to remove the noisecorrelation or whiten the noise to improve system recognition andrejection performance. In this case the whitening processing is:H_(q)=Noise_Correlation_Matrix^((1/2))

Another case would be for decoder 2004 to apply a “phase-only” orall-pass filter prior to forming the iris feature code. A phase-only andall-pass filter has a unit magnitude frequency response and non-zerophase response. This type of filter is equivalent to spatially shiftingdifferent spatial frequency components by different amounts, but leavingthe magnitude of the different spatial frequency components unchanged.Application of this type of filtering in decoder 2004 would not changethe power spectrum of the additive noise n(x) and hence not correlatethe additive noise n(x).

Another case would be for decoder 2004 to apply an all-pass filter tocorrect the phase of the different spatial frequency components of thesignal while also multiplicatively modifying the amplitude of thespatial frequency components with values close to (including less than)one. This would yield a minimum of noise amplification and possibly areduction of noise power. Changing the amplitude of the spatialfrequency components would change the spatial correlation of theadditive noise; this change may be balanced with a decrease in additivenoise power for a particular iris feature code forming function.

The optics of wavefront coded imaging systems can be selected anddesigned so as to maximize certain types of image information transferas well as to yield imaging advantages such as large depth of field,insensitivity to optical and mechanical aberrations and aliasing, etc.The information content of wavefront coded images can be considered as afunction of spatial frequency. All practical images have noise. Thisnoise acts to reduce the information content of the images. If the noisehas essentially the same amount of RMS power at each spatial frequency,then the noise affects the information as a function of spatialfrequency equally. The MTF of the imaging system varies as a function ofspatial frequency. As information is closely related to signal-to-noiseratios, a spatial frequency component of an image formed with a highvalued MTF has a higher information value than if formed with a lowervalued MTF (assuming the same RMS noise power). In terms of the Hammingdistance, two iris feature codes of the same specialized iris thatcontains only a single spatial frequency component will statisticallyincrease in Hamming distance (become less similar) as the MTF value atthe specific spatial frequency decreases. The Hamming distance will alsostatistically decrease (become more similar) for two different yetspecialized irises as the MTF value at the specific spatial frequencydecreases.

Rectangularly separable wavefront coding optics allows a high degree ofinformation transfer in the x-y plane. If information transfer should bemore angularly independent, if for example the angular orientation ofthe iris when imaged is not closely controlled, then non-separableoptics should be used. The MTFs from these non-separable optics shouldbe more circularly symmetric than is possible with rectangularlyseparable optics. Circularly symmetric wavefront coding optics can alsobe used in a case where the optical form is composed of the weighted sumof polynomials in the radius variable. Constant profile path optics arealso useful for these systems, as are linear combinations of cosineterms in the form:P(r,theta)=Σa _(i) r ^(i) cos (w _(i)theta+phi_(i))

Since certain changes may be made in the above methods and systemswithout departing from the scope thereof, it is intended that all mattercontained in the above description or shown in the accompanying drawingbe interpreted as illustrative and not in a limiting sense. It is alsoto be understood that the following claims are to cover certain genericand specific features described herein.

1. A biometric optical recognition system, comprising: optics, includinga wavefront coding element, for imaging a wavefront of an object to berecognized to an intermediate image; and a detector for detecting theintermediate image, wherein a modulation transfer function detected bythe detector contains no zeros such that subsequent task based imageprocessing recognizes the object.
 2. The system of claim 1, furthercomprising a decoder, connected with the detector, for implementing thetask based image processing.
 3. The system of claim 2, the decoderoperable as an all-pass filter in the frequency domain.
 4. The system ofclaim 2, the decoder operable as an attenuation filter in the frequencydomain for magnifications of one or less.